Abstract

A method is presented whereby any photometer than can measure relative diffuse reflectance of materials in the laboratory can be used to measure absolute diffuse reflectance of such samples without any modification to the apparatus. No additional integrating spheres or special optical geometries are required. All that is needed are two powdered materials, one a very strong absorber and one a very weak absorber at the wavelength of interest, a requirement that is not difficult to meet. A theoretical development based on the Kubelka-Munk theory, the results of experimental testing, and an error analysis are presented.

© 1987 Optical Society of America

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References

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  1. J. A. Van den Akker, L. R. Dearth, W. M. Shilcox, “Evaluation of Absolute Reflectance for Standardization Purposes,” J. Opt. Soc. Am. 56, 250 (1966).
    [CrossRef]
  2. D. G. Goebel, B. P. Caldwell, H. K. Hammond, “Use of an Auxillary Sphere with a Spectroreflectometer to Obtain Absolute Reflectance,” J. Opt. Soc. Am. 56, 783 (1966).
    [CrossRef]
  3. W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand. Sect. A 80, 585 (1976).
    [CrossRef]
  4. F. Grum, T. E. Wightman, “Absolute Reflectance of Eastern White Reflectance Standard,” Appl. Opt. 16, 2775 (1977).
    [CrossRef] [PubMed]
  5. W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).
  6. G. Kortum, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
    [CrossRef]
  7. G. Kortum, D. Oelkrug, “The Scattering Coefficients of the Kubelka-Munk Theory,” Z. Naturforsch. Teil A 19, 28 (1964).
  8. It is conventional to use a subscript of infinity with R to indicate reflectance from a powdered sample that is so thick that all incident radiation is either reflected or absorbed. In practice for the samples and materials used in this work a few millimeters are sufficient to be infinitely thick. Since all reflectance values involved in this report are for such infinitely thick samples, the subscript has been dropped.
  9. P. S. Mudgett, L. W. Richards, “Multiple Scattering Calculations for Technology,” Appl. Opt. 10, 1485 (1971).
    [CrossRef] [PubMed]
  10. B. J. Brinkworth, “Interpretation of the Kubelka-Munk Coefficients in Reflection Theory,” Appl. Opt. 11, 1434 (1972).
    [CrossRef] [PubMed]
  11. Since, as is pointed out later, a multiplicative constant cancels out of the concentration, the value of concentration may actually be expressed in any convenient units.
  12. S. Hedelman, W. N. Mitchell, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 158.
    [CrossRef]

1977 (1)

1976 (1)

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand. Sect. A 80, 585 (1976).
[CrossRef]

1972 (1)

1971 (1)

1966 (2)

1964 (1)

G. Kortum, D. Oelkrug, “The Scattering Coefficients of the Kubelka-Munk Theory,” Z. Naturforsch. Teil A 19, 28 (1964).

Brinkworth, B. J.

Budde, W.

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand. Sect. A 80, 585 (1976).
[CrossRef]

Caldwell, B. P.

Dearth, L. R.

Goebel, D. G.

Grum, F.

Hammond, H. K.

Hecht, H. G.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

Hedelman, S.

S. Hedelman, W. N. Mitchell, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 158.
[CrossRef]

Kortum, G.

G. Kortum, D. Oelkrug, “The Scattering Coefficients of the Kubelka-Munk Theory,” Z. Naturforsch. Teil A 19, 28 (1964).

G. Kortum, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

Mitchell, W. N.

S. Hedelman, W. N. Mitchell, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 158.
[CrossRef]

Mudgett, P. S.

Oelkrug, D.

G. Kortum, D. Oelkrug, “The Scattering Coefficients of the Kubelka-Munk Theory,” Z. Naturforsch. Teil A 19, 28 (1964).

Richards, L. W.

Shilcox, W. M.

Van den Akker, J. A.

Wendlandt, W. W.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

Wightman, T. E.

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

J. Res. Natl. Bur. Stand. Sect. A (1)

W. Budde, “Calibration of Reflectance Standards,” J. Res. Natl. Bur. Stand. Sect. A 80, 585 (1976).
[CrossRef]

Z. Naturforsch. Teil A (1)

G. Kortum, D. Oelkrug, “The Scattering Coefficients of the Kubelka-Munk Theory,” Z. Naturforsch. Teil A 19, 28 (1964).

Other (5)

It is conventional to use a subscript of infinity with R to indicate reflectance from a powdered sample that is so thick that all incident radiation is either reflected or absorbed. In practice for the samples and materials used in this work a few millimeters are sufficient to be infinitely thick. Since all reflectance values involved in this report are for such infinitely thick samples, the subscript has been dropped.

W. W. Wendlandt, H. G. Hecht, Reflectance Spectroscopy (Interscience, New York, 1966).

G. Kortum, Reflectance Spectroscopy (Springer-Verlag, New York, 1969).
[CrossRef]

Since, as is pointed out later, a multiplicative constant cancels out of the concentration, the value of concentration may actually be expressed in any convenient units.

S. Hedelman, W. N. Mitchell, in Modern Aspects of Reflectance Spectroscopy, W. W. Wendlandt, Ed. (Plenum, New York, 1968), p. 158.
[CrossRef]

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Tables (3)

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Table I List of Symbols

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Table II Absolute Reflectance of Colored Paper

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Table III Absolute Reflectance of BaSO4 Standard

Equations (25)

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R = M s / M b ,
R = δ R .
F ( R ) = ( 1 - R ) 2 2 R = k s .
k = A C ,
F ( δ R ) = ( 1 - δ R ) 2 2 δ R = A s C ,
d [ log F ( R ) ] d ( log C ) = 1 ,
log F ( β R ) = log ( 1 - β R ) 2 2 β R
d [ log F ( β R ) ] d ( log C ) = d [ log F ( β R ) ] d R d R d ( log C ) .
C = s A ( 1 - δ R ) 2 2 δ R .
d [ log F ( β R ) ] d ( log C ) = ( 1 + β R ) ( 1 - δ R ) ( 1 - β R ) ( 1 + δ R ) .
d 2 [ log F ( β R ) ] d 2 ( log C ) = R log e [ ( 1 + β R ) ( 1 - δ R ) ( 1 - β R ) 2 ( 1 + δ R ) 2 + ( 1 - δ R ) 2 ( 1 - β R ) ( 1 + δ R ) 3 ] ( δ - β ) .
log F ( β R 1 ) - log F ( β R 2 ) log C 1 - log C 2 = 1.
β = δ = - R 1 R 2 ( C 1 - C 2 ) - ( R 1 - R 2 ) ( C 1 C 2 R 1 R 2 ) 1 / 2 R 1 R 2 ( C 2 R 1 - C 1 R 2 ) .
β R 1 = ( 1 + β R 1 ) ( 1 - β R 2 ) 2 R 1 ( R 2 - R 1 ) ,
β R 2 = - ( 1 + β R 2 ) ( 1 - β R 1 ) 2 R 2 ( R 2 - R 1 ) ,
β C 1 = - ( 1 + β R 1 ) ( 1 - β R 2 ) 2 C 1 ( R 2 - R 1 ) ,
β C 2 = - ( 1 - β R 1 ) ( 1 - β R 2 ) 2 C 2 ( R 2 - R 1 ) .
( σ β ) 2 = [ ( β R 1 ) 2 + ( β R 2 ) 2 ] ( σ R ) 2 + ( β C 1 ) 2 ( σ C 1 ) 2 + ( β C 2 ) 2 ( σ C 2 ) 2 ,
C = 1 V ( f W p W p + W w ) ( W g - W d ) ,
f = W b p W w p + W b p .
C W p = f ( W g - W d ) V [ 1 ( W p + W w ) - W p ( W p + W w ) - 2 ] ,
C W w = - f W p V ( W g - W d ) ( W p + W w ) - 2 ,
C W g = f W p V ( W p + W w ) ,
C W d = - f W p V ( W P + W w ) .
σ C = [ ( C W p ) 2 ( σ W p ) 2 + ( C W w ) 2 ( σ W w ) 2 + ( C W g ) 2 ( σ W g ) 2 + ( C W d ) 2 ( σ W d ) 2 ] 1 / 2 .

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